|
| |
|
|
A143409
|
|
Square array read by antidiagonals: form the Euler-Seidel matrix for the sequence {k!} and then divide column k by k!.
|
|
4
| |
|
|
1, 2, 1, 5, 3, 1, 16, 11, 4, 1, 65, 49, 19, 5, 1, 326, 261, 106, 29, 6, 1, 1957, 1631, 685, 193, 41, 7, 1, 13700, 11743, 5056, 1457, 316, 55, 8, 1, 109601, 95901, 42079, 12341, 2721, 481, 71, 9, 1, 986410, 876809, 390454, 116125, 25946, 4645, 694, 89, 10, 1
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| The Euler-Seidel matrix for the sequence {k!} is array A076571 read as a square, whose k_th column entries have a common factor of k!. Removing these common factors gives the current table. This table is closely connected to the constant 1/e. The row, column and diagonal entries of this table occur in series acceleration formulas for 1/e. For a similar table based on the differences of the sequence {k!} and related to the constant e, see A086764. For other arrays similarly related to constants see A143410 (for sqrt(e)), A143411 (for 1/sqrt(e)), A008288 (for log(2)), A108625 (for zeta(2)) and A143007 (for zeta(3)).
|
|
|
LINKS
| D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
Eric Weisstein's World of Mathematics Poisson-Charlier polynomial
|
|
|
FORMULA
| T(n,k) = 1/k!*sum {j = 0..n} C(n,j)*(k+j)!. T(n,k) = (n+k)!/k!*Num_Pade(n,k), where Num_Pade(n,k) denotes the numerator of the Pade approximation for the function exp(x) of degree (n,k) evaluated at x = 1. Recurrence relations: T(n,k) = T(n-1,k) + (k+1)*T(n-1,k+1); T(n,k) = (n+k)*T(n-1,k) + T(n-1,k-1). E.g.f for column k: exp(y)/(1-y)^(k+1). E.g.f. for array: exp(y)/(1-x-y) = (1 + x + x^2 + ...) + (2 + 3*x + 4*x^2 + ...)*y + (5 + 11*x + 19*x^2 + ...)*y^2/2! + ... . Row n lists the values of the Poisson-Charlier polynomial x^(n) + C(n,1)*x^(n-1) + C(n,2)*x^(n-2) + ... + C(n,n) for x = 1,2,3,..., where x^(m) denotes the rising factorial x*(x+1)*...*(x+m-1). Main diagonal is A001517. Series formulas for 1/e: Row n: 1/e = n!*[1/T(n,0) - 1/(1!*T(n,0)*T(n,1)) + 1/(2!*T(n,1)*T(n,2)) - 1/(3!*T(n,2)*T(n,3)) + ...]. Column k: k!/e = A000166(k) + (-1)^(k+1)*[0!/(T(0,k)*T(1,k)) + 1!/(T(1,k)*T(2,k)) + 2!/(T(2,k)*T(3,k)) + ...]. Main diagonal: 1/e = 1 - 2* sum {n = 0..inf} (-1)^n/(T(n,n)*T(n+1,n+1)) = 1 - 2*[1/(1*3) - 1/(3*19) + 1/(19*193) - ...]. Second subdiagonal: 1/e = 2*[1^2/(1*5) - 2^2/(5*49) + 3^2/(49*685) - ...]. Compare with A143413.
|
|
|
EXAMPLE
| The Euler-Seidel matrix for the sequence {k!} begins
==============================================
n\k|.....0.....1.....2.....3.....4.....5.....6
==============================================
0..|.....1.....1.....2.....6....24...120...720
1..|.....2.....3.....8....30...144...840
2..|.....5....11....38...174...984
3..|....16....49...212..1158
4..|....65...261..1370
5..|...326..1631
6..|..1957
...
Dividing the k_th column by k! gives
==============================================
n\k|.....0.....1.....2.....3.....4.....5.....6
==============================================
0..|.....1.....1.....1.....1.....1.....1.....1
1..|.....2.....3.....4.....5.....6.....7
2..|.....5....11....19....29....41
3..|....16....49...106...193
4..|....65...261...685
5..|...326..1631
6..|..1957
...
Examples of series formula for 1/e:
Row 2: 1/e = 2*[1/5 - 1/(1!*5*11) + 1/(2!*11*19) - 1/(3!*19*29) + ...].
Column 4: 24/e = 9 - [0!/(1*6) + 1!/(6*41) + 2!/(41*316) + ...].
|
|
|
MAPLE
| with combinat: T := (n, k) -> 1/k!*add(binomial(n, j)*(k+j)!, j = 0..n): for n from 0 to 9 do seq(T(n, k), k = 0..9) end do;
|
|
|
CROSSREFS
| Cf. A008288, A076571, A086764, A108625, A143007, A143410, A143411, A143413, A001517 (main diagonal), A028387 (row 2), A000522 (column 0), A001339 (column 1), A082030 (column 2), A095000 (column 3), A095177 (column 4).
Sequence in context: A188416 A160185 A188392 * A197387 A171177 A171176
Adjacent sequences: A143406 A143407 A143408 * A143410 A143411 A143412
|
|
|
KEYWORD
| easy,nonn,tabl
|
|
|
AUTHOR
| Peter Bala (pbala(AT)toucansurf.com), Aug 14 2008
|
| |
|
|
